La profesora explica la idea con paciencia.

Because the subject is a woman. Spanish job titles usually have masculine and feminine forms, and the definite article must agree:

• Masculine: el profesor
• Feminine: la profesora Colloquially, you might also hear la profe in Spain.

Yes, but it changes the meaning:

• la profesora = the (specific/known) teacher
• una profesora = a (non-specific) teacher Spanish usually needs an article with singular count nouns; you don’t drop it as English sometimes does.

Spanish uses the so‑called “personal a” before direct objects that are people (or personified), not things. Since la idea is not a person, you say explica la idea, not explica a la idea.
Compare:

• Explica la idea a los alumnos. (She explains the idea to the students.)
  Here, the a introduces the indirect object (the recipients), not the idea.

Yes: La profesora está explicando la idea con paciencia.

• explica can mean “is explaining” (right now) or “explains” (habitually).
• está explicando emphasizes the ongoing action happening now.

Yes. Common options:

• La profesora explica la idea con paciencia. (neutral)
• La profesora explica con paciencia la idea. (slight emphasis on the manner)
• Con paciencia, la profesora explica la idea. (fronted for emphasis; add a comma) All are correct; choose based on what you want to highlight.

Use an indirect object (often with a pronoun):

• La profesora les explica la idea a los alumnos.
  Indirect object pronouns: me, te, le, nos, os, les. In Spanish it’s normal to use both the pronoun and the a + [person] phrase for clarity.

Use the direct object pronoun la (fem. sing.):

• La profesora la explica con paciencia.
  If you also include an indirect object pronoun (e.g., to the students), remember le/les + la → se la:
• La profesora se la explica a los alumnos.

Because explicar takes a direct object: explicar algo.
Use:

• explicar la idea (correct)
  But with other verbs:
• hablar de la idea (talk about the idea)
• dar una explicación sobre la idea (give an explanation about the idea)

Approximate Castilian pronunciation:

• La profesora: la pro-fe-SO-ra
• explica: eks-PLI-ka (the x is like “ks”)
• la idea: la i-DE-a (the d between vowels is soft, like English “th” in “this”)
• con paciencia: kon pa-THYEN-thya (the c before i/e is a “θ” sound in most of Spain; in Latin America it’s an “s”: pa-SYEN-sya)

Singular count nouns in Spanish usually need an article (definite or indefinite). So you say:

• explica la idea (the idea)
• explica una idea (an idea)
  Leaving it out would sound ungrammatical here.

By itself, explica can be either:

• Right now: “is explaining”
• Habitual: “(she) explains”
  Context or time markers clarify it: Ahora la profesora explica… / Siempre la profesora explica…